The general principles of operation for a plasma centrifuge are well known and well understood. In short, a plasma centrifuge generates forces on charged particles which will cause the particles to separate from each other according to their mass. More specifically, a plasma centrifuge relies on the effect crossed electric and magnetic fields have on charged particles. As is known, crossed electric and magnetic fields will cause charged particles in a plasma to move through the centrifuge on respective helical paths around a centrally oriented longitudinal axis. As the charged particles transit the centrifuge under the influence of these crossed electric and magnetic fields they are, of course, subject to various forces. Specifically, in the radial direction, i.e. a direction perpendicular to the axis of particle rotation in the centrifuge, these forces are: 1) a centrifugal force, F.sub.c, which is caused by the motion of the particle; :2) an electric force, F.sub.E, which is exerted on the particle by the electric field, E.sub.r ; and 3) a magnetic force, F.sub.B, which is exerted on the particle by the magnetic field, B.sub.z. Mathematically, the magnitudes of each of these forces are respectively expressed as:
F.sub.c =Mr.omega..sup.2 ; PA1 F.sub.E =eE.sub.r ; and PA1 F.sub.B =er.omega.B.sub.z PA1 M is the mass of the particle; PA1 r is the distance of the particle from its axis of rotation; PA1 .omega. is the angular frequency of the particle; PA1 e is the electric charge of the particle; PA1 E is the electric field strength; and PA1 M.sub.C =e(B.sub.z a).sup.2 /(8V.sub.ctr) where a is the radius of the chamber. We also used: EQU .alpha.=(.PHI..sub.o -.PHI.)/(.omega.-.omega..sub.o)=2V.sub.ctr /(a.sup.2 B.sub.z) (Eq. 2)
Where:
B.sub.z is the magnetic flux density of the field.
In a plasma centrifuge, it is universally accepted that the electric field will be directed radially inward. Stated differently, there is an increase in positive voltage with increased distance from the axis of rotation in the centrifuge. Under these conditions, the electric force F.sub.E will oppose the centrifugal force F.sub.c acting on the particle, and the magnetic force resulting from the E.times.B rotation aids the outward centrifugal forces. Accordingly, an equilibrium condition in a radial direction of the centrifuge can be expressed as: EQU .SIGMA.F.sub.r =0 (positive direction radially outward) EQU F.sub.c -F.sub.E +F.sub.B =0 EQU Mr.omega..sup.2 -eE.sub.r +er.omega.B.sub.z =0 (Eq. 1)
It is noted that Eq. 1 has two real solutions, one positive and one negative, namely: ##EQU1##
where .OMEGA.=eB.sub.z /M.sub.c is the cyclotron frequency of an ion with mass M.
For a plasma centrifuge, the intent is to seek an equilibrium to create conditions in the centrifuge which allow the centrifugal forces, F.sub.c, to separate the particles from each other according to their mass. This happens because the centrifugal forces differ from particle to particle, according to the mass (M) of the particular particle. Thus, particles of heavier mass experience greater Fc and move more toward the outside edge of the centrifuge than do the lighter mass particles which experience smaller centrifugal forces. The result is a distribution of lighter to heavier particles in a direction outward from the mutual axis of rotation. As is well known, however, a plasma centrifuge will not completely separate all of the particles in the aforementioned manner.
As indicated above in connection with Eq. 1, a force balance can be achieved for all conditions when the electric field E is chosen to confine ions, and ions exhibit confined orbits. In the plasma filter of the present invention, unlike a centrifuge, the electric field is chosen with the opposite sign to extract ions. The result is that ions of mass greater than a cut-off value, M.sub.c, are on unconfined orbits. The cut-off mass, M.sub.c, can be selected by adjusting the strength of the electric and magnetic fields. The basic features of the plasma filter can be described using the Hamiltonian formalism.
The total energy (potential plus kinetic) is a constant of the motion and is expressed by the Hamiltonian operator: EQU H=e.PHI.+(P.sub.r.sup.2 +P.sub.z.sup.2)/(2M)+(P.sub..theta. -e.PSI.).sup.2 /(2Mr.sup.2)
where P.sub.r =Mv.sub.r, P.sub..theta. =Mrv.sub..theta. +e.PSI., and P.sub.z =Mv.sub.z are the respective components of the momentum and e.PHI. is the potential energy. .omega.=r.sup.2 B.sub.z /2 is related to the magnetic flux function and .PHI.=V.sub.ctr -.alpha..omega. is the electric potential. E=-.gradient..PHI. is the electric field which is chosen to be greater than zero for the filter case of interest. We can rewrite the Hamiltonian: EQU H=eV.sub.ctr -e.alpha.r.sup.2 B.sub.z /2+(P.sub.r.sup.2 +P.sub.z.sup.2)/(2M) +(P.sub..theta. -er.sup.2 B.sub.z /2).sup.2 /(2Mr.sup.2)
We assume that the parameters are not changing along the z axis, so both P.sub.z and P.sub..theta. are constants of the motion. Expanding and regrouping to put all of the constant terms on the left hand side gives: EQU H-eV.sub.ctr -P.sub.z.sup.2 /(2M)+P.sub..theta..OMEGA./2=P.sub.r.sup.2 /(2M)+(P.sub..theta..sup.2 /(2Mr.sup.2)+(M.OMEGA.r.sup.2 /2)(.OMEGA./4-.alpha.)
where .OMEGA.=e B.sub.z /M is the cyclotron frequency.
The last term is proportional to r.sup.2, so if .OMEGA./4-.alpha.&lt;0 then, since the second term decreases as 1/r.sup.2, P.sub.r.sup.2 must increase to keep the left-hand side constant as the particle moves out in radius. This leads to unconfined orbits for masses greater than the cut-off mass given by:
where .PHI..sub.0 =V.sub.ctr and .omega..sub.o =0 while at r=a, .phi.=0 and .omega.=a.sup.2 B.sub.z /2
So, for example, normalizing to the proton mass, M.sub.p, we can rewrite Eq. 2 to give the voltage required to put higher masses on loss orbits: EQU V.sub.ctr &lt;1.2.times.10.sup.-1 (a(m)B.sub.z (gauss)).sup.2 /(M.sub.C /M.sub.P)
Hence, a device radius of 1 m, a cutoff mass ratio of 100, and a magnetic field of 200 gauss require a voltage of 48 volts.
The same result for the cut-off mass in the plasma mass filter can again be obtained by looking at the simple force balance equation given by: EQU .SIGMA.F.sub.r =0 (positive direction radially outward) EQU F.sub.c +F.sub.E +F.sub.B =0 EQU Mr.omega..sup.2 +eEr-er.omega.B.sub.z =0 (Eq. 3)
which differs from Eq. 1 by the signs of the electric field and magnetic forces and has the solutions: ##EQU2##
so if 4E.sub.r /r.OMEGA.B.sub.z &gt;1 then .omega. has imaginary roots and the force balance cannot be achieved. For a filter device with a cylinder radius "a", a central voltage, V.sub.ctr, and zero voltage on the wall, the same expression for the cut-off mass is found to be: EQU M.sub.C =ea.sup.2 B.sub.z.sup.2 /(8 V.sub.ctr)
and when the mass M of a charged particle is greater than the threshold value (M&gt;M.sub.c), the particle will continue to move radially outwardly until it strikes the wall, whereas the lighter mass particles will be contained and can be collected at the exit of the device. The higher mass particles can also be recovered from the walls using various approaches.
It is important to note that for a given device the value for M.sub.c in equation 3 is determined by the magnitude of the magnetic field, B.sub.z, and the voltage at the center of the chamber (i.e. along the longitudinal axis), V.sub.ctr. These two variables are design considerations and can be controlled.
To more fully appreciate the consequences of the mathematical computations set forth above, a comparison of the physics involved in the respective operations of a plasma mass filter and a plasma centrifuge is instructive. Although there are similarities between the two types of devices, these similarities are, for the most part, superficial. For instance, both types of devices establish an axially oriented magnetic field. Both establish crossed electric and magnetic fields (albeit the directions of the electric fields would be radially opposite to each other). And, both are intended to separate charged particles in a multi-species plasma from each other according to their mass. The similarities, however, end there.
An important distinction between plasma mass filters and plasma centrifuges is the fact that a plasma centrifuge operationally relies on collisions between the various charged particles in the plasma. Specifically, it is the collisions between light and heavy ions in a centrifuge that establish the operative mechanism for separating particles according to their mass. A plasma mass filter, on the other hand, does not use this collisional mechanism for its operation. In fact, to the contrary, a plasma mass filter relies on the avoidance of collisions between charged particles in the plasma. The purpose for doing this in a plasma mass filter is to thereby allow each charged particle to follow a predetermined trajectory. It then follows that the separation of charged particles in a plasma mass filter is possible because the respective trajectories of the particles differ according to the mass of the particular charged particle. This basic distinction leads to still other distinctions between a plasma mass filter and a plasma centrifuge.
Two major differences should be noted between the operational regimes of the centrifuge and the mass filter. First, the radial electric field in the conventional centrifuge is oriented inward to confine all of the ions. In terms of individual ion orbits, this electric field is the only radial force balancing the outwardly-directed centrifugal and vxB forces. In the filter, the electric field is oriented outward to extract ions. For masses below the cutoff mass the now inwardly-directed vxB force can balance the outwardly-directed electric and centrifugal forces to achieve radial confinement. For masses above the cutoff mass, however, the inwardly-directed vxB force is insufficient to balance the outwardly-directed electric and centrifugal forces and these ions are expelled.
The above orbit comparison ignores the effects of ion-ion collisions, which is the source of the second major difference between the filter and the centrifuge. The filter operates in a regime where the collisions are infrequent so that the trajectories are fundamentally those given by the balance of centrifugal, vxB, and electric forces; separation results primarily from the radial expulsion of the heavy particles with mass in excess of the cutoff mass. By contrast, the centrifuge achieves its more limited mass separation through collisions which drive the various ion species to a thermodynamic equilibrium state. In this equilibrium state, the ratio of the radial distributions of the light and heavy ion densities is a Gaussian whose half-width depends on the difference in the centrifugal forces between the heavy and light ions.
It is well known that the frequency of collisions between charged particles in a multi-species plasma is proportional to the density of the plasma. Furthermore, it is known that if a charged particle is able to avoid a collision with another particle, the unobstructed particle can be influenced by crossed electric and magnetic fields to follow a predetermined trajectory. Specifically, it is known that any particle having a charge, e, and a velocity, v, perpendicular to a magnetic field of intensity B will move along a circular path. Under these conditions, the number of revolutions the particle makes around this circular path per second is known as the cyclotron frequency, and can be mathematically expressed as; .OMEGA.=B.sub.z e/m, where m is the mass of the particle. It is important to note that, in line with the mathematics set forth above, the cyclotron frequency is independent of the velocity, v, of the particle.
As indicated above, it is an important concept for the operation of a plasma mass filter that the charged particles be able to avoid collisions with other charged particles in the plasma. Although, there can be no absolute assurances that collisions can be completely avoided, it is clear that the probability of collisions can be reduced simply by reducing the density of the plasma in the chamber. With this in mind, and for purposes of the present invention, a "collisional density" is defined as being the plasma density below which the mathematics disclosed above for the determination of M.sub.c are effective for describing the operation of the plasma mass filter. Stated differently, in a continuum, the "collisional density" is a transition point between the higher plasma densities which are useful for the operation of a plasma centrifuge and the lower plasma densities which are useful for the operation of a plasma mass filter. As a practical matter, a plasma mass filter is effective at densities below a "collisional density" wherein the ratio of the cyclotron frequency of particles to the collisional frequency of the particles is greater than approximately one.
To further contrast the operation of a plasma mass filter with the operation of a plasma centrifuge, it is to be appreciated that the trajectories of the lighter charged particles are established in a plasma mass filter so that these particles will exit the filter chamber for subsequent collection. Also, in order to maintain a charge balance in the filter chamber, electrons are removed from the chamber, as necessary, along with the lighter charged particles. On the other hand, because the trajectories of the heavier charged particles are different than the trajectories of the lighter charged particles, the heavier charged particles can be directed into contact with a collector which is located where the light charged particles do not travel. A plasma centrifuge has no such reliance on differences in trajectories. Instead, as discussed above, the centrifuge relies on collisions between particles which effectively disrupt their otherwise predictable trajectories.
In light of the above it is an object of the present invention to provide a plasma mass filter which effectively separates low-mass charged particles from high-mass charged particles. It is another object of the present invention to provide a plasma mass filter which has variable design parameters which permit the operator to select a demarcation between low-mass particles and high-mass particles. Yet another object of the present invention is to provide a plasma mass filter which is easy to use, relatively simple to manufacture, and comparatively cost effective.